Think about what C(n-k,k) looks like in Pascal’s triangle. The sums you have are essentially (barring a leading term C(n,0)) sums along diagonals of Pascal’s triangle

Like I wrote, the sum asked about is essentially Fibonacci numbers. However it’s missing the leading term C(n,0). Without the leading term the sequence is just Fibonacci numbers minus one: 0,0,1,2,4,7,12,\ldots.

Compare (python code, implement choose yourself)

[sum(choose(n-k,k) for k in range(1,n//2+1)) for n in range(0,10)]
-> [0, 0, 1, 2, 4, 7, 12, 20, 33, 54]

with

[sum(choose(n-k,k) for k in range(0,n//2+1)) for n in range(0,10)]
-> [1, 1, 2, 3, 5, 8, 13, 21, 34, 55]